164. The difficulties in the way of an exact mathematical treatment of diffusion are similar to those which occurred in the problems of viscosity and heat conduction. Following the procedure we adopted in discussing these earlier problems, we shall begin by giving a simple, but mathematically inexact, treatment of the question.

We imagine two gases diffusing through one another in a direction parallel to the axis of z, the motion being the same at all points in a plane perpendicular to the axis of z. The gases are accordingly arranged in layers perpendicular to this axis.

The simplest case arises when the molecules of the two gases are similar in mass and size–like the red and white billiard balls we discussed in § 6. In other cases differences in the mass and size of the molecules tend, as the motion of the molecules proceeds, to set up differences of pressure in the gas. The gas adjusts itself against these by a slow mass-motion, which will of course be along the axis of z at every point.

Let us denote the mass-velocity in the direction of z increasing by w0, and let the molecular densities of the two gases be ν1, ν2. Then ν1, ν2 and w0 are functions of z only.